Square-free Order Research Articles

Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order

Abstract In this article, we study the extent to which an n-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an n-dimensional compact flat manifold with the cyclic holonomy group of square-free order.

Finite skew braces of square-free order and supersolubility

Abstract The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if $(B,+)$ is nilpotent. Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

Nowhere-zero 3-flows in Cayley graphs on supersolvable groups

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

CI-property of [formula omitted] and [formula omitted] for digraphs

We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square-free order are DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).

On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq

Abstract A graph is said to be symmetric if its automorphism group is transitive on its arcs. This article is one of a series of articles devoted to characterizing prime-valent arc-transitive graphs of square-free order or twice square-free order. In this article, we determine all 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq.

Fuglede’s Conjecture Holds for Cyclic Groups of Order pqrs

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Łaba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most three primes as the order of the cyclic group was studied intensely in the last couple of years. In this paper we study the case when the order of the cyclic group is the product of four different primes and prove that Fuglede’s conjecture holds in this case.

The prime graphs of some classes of finite groups

In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of order $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any $n\in \mathbb{N}$, solvable groups of $n^\text{th}$-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or $A_5$ and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.

Arc-Transitive Graphs of Square-free Order with Valency 11

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products

Let L/K be a G-Galois extension of fields with an H-Hopf Galois structure of type N. We study the Galois correspondence ratio GC(G, N), which is the proportion of intermediate fields E with K ⊆ E ⊆ L that are in the image of the Galois correspondence for the H-Hopf Galois structure on L/K. The Galois correspondence ratio for a Hopf Galois structure can be found by translating the problem into counting certain subgroups of the corresponding skew brace. We look at skew braces arising from finite radical algebras A and from Zappa–Sz´ep products of finite groups, and in particular when A3 = 0 or the Zappa–Sz´ep product is a semidirect product, in which cases the corresponding skew brace is a bi-skew brace, that is, a set G with two group operations ◦ and ? in such a way that G is a skew brace with either group structure acting as the additive group of the skew brace. We obtain the Galois correspondence ratio for several examples. In particular, if (G, ◦, ?) is a biskew brace of squarefree order 2m where (G, ◦) ∼= Z2m is cyclic and (G, ?) ∼= Dm is dihedral, then for large m, GC(Z2m, Dm) is close to 1/2 while GC(Dm, Z2m) is near 0.

The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups

In the recent past, results have shown that Nilpotent groups such as p-groups, have normal series of finite length. Any finite p-group has many normal subgroups and consequently, the phenomenon of large number of non-isomorphic subgroups of a given order. This makes it an ideal object for combinatorial and cohomological investigations. Cartesian product (otherwise known as the product set) plays vital roles in the course of synthesizing the abstract groups. Previous studies have determined the number of distinct fuzzy subgroups of various finite p-groups including those of square-free order. However, not much work has been done on the fuzzy subgroup classification for the nilpotent groups formed from the Cartesian products of p-groups through their computations. Here, part of our intention is therefore trying to make some designs so as to classify the nilpotent groups formed from the Cartesian products of p-groups through their computations. The Cartesian products of p-groups were taken to obtain nilpotent groups. Results up to two dimensions are now obtainable. In this paper, the fuzzy subgroups of the nilpotent product of two abelian subgroups of orders 2^<i>n ^</i>and 128. The integers n ≥ 7 have been successfully considered and the derivation for the explicit formulae for its number distinct fuzzy subgroups were calculated. Some methods were once being used in counting the chains of fuzzy subgroups of an arbitrary finite <i>p</i>-group <I>G. </I>Here, the adoption of the famous Inclusion-Exclusion principle is very necessary and imperative so as to obtain a reasonable, and as much as possible accurate.

Generation of finite groups with cyclic Sylow subgroups

Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. We generalise Slattery’s result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.

On a Divisibility Property Involving the Sum of Element Orders

A finite group G is called $$\psi $$ -divisible if $$\psi (H)|\psi (G)$$ for any subgroup H of G, where $$\psi (H)$$ and $$\psi (G)$$ are the sum of element orders of H and G, respectively. In this paper, we classify the finite groups whose subgroups are all $$\psi $$ -divisible. Since the existence of $$\psi $$ -divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $$\psi $$ -divisibility property of ZM-groups. In the end, we introduce the concept of $$\psi $$ -normal divisible group, i.e., a group for which the $$\psi $$ -divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $$\psi $$ -normal divisible groups which are neither simple nor nilpotent.

Skew braces of squarefree order

Let [Formula: see text] be a squarefree integer, and let [Formula: see text], [Formula: see text] be two groups of order [Formula: see text]. Using our previous results on the enumeration of Hopf–Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group [Formula: see text] and additive group [Formula: see text]. As an application, we enumerate skew braces whose order is the product of three distinct primes, in particular proving a conjecture of Bardakov, Neshchadim and Yadav on the number of skew braces of order [Formula: see text] for primes [Formula: see text].

On the multiple holomorph of groups of squarefree or odd prime power order

Let G be a group and write Perm(G) for its symmetric group. Let us define Hol(G) to be the holomorph of G, regarded as a subgroup of Perm(G), and let NHol(G) denote its normalizer. The quotient T(G)=NHol(G)/Hol(G) has been computed for various families of groups G, and in most of the known cases, it turns out to be elementary 2-abelian, except for two groups of order 16, and some groups of odd prime power order and nilpotency class two. In this paper, we shall show that T(G) is elementary 2-abelian for all finite groups G of squarefree order, and that T(G) is not a 2-group for certain finite p-groups G of nilpotency class at most p−1.

S-Arc-regular prime-valent Cayley graphs of square-free order

A high cited paper [Feng and Li, One-regular graphs of square-free order of prime valency, Europ. J. Combin. 32 (2011) 261–275] classified 1-arc-regular prime-valent graphs (necessarily Cayley graphs) of square-free order. It is thus interesting to ask what are the [Formula: see text]-arc-regular prime-valent Cayley graphs of square-free order for each [Formula: see text]. In this paper, a complete list of all such graphs is given. The proving process also involves a complete list of the vertex stabilizers of [Formula: see text]-arc-regular prime-valent graphs of any order.

Locally-primitive Arc-transitive 10-Valent Graphs of Square-free Order

In this paper, we present a complete classification for locally-primitive arc-transitive graphs which have square-free order and valency 10. The classification involves nine graphs and three infinite families of graphs.

A family of edge-transitive Cayley graphs

Edge-transitive graphs of order a prime or a product of two distinct primes with any positive integer valency, and of square-free order with valency at most 7 have been classified by a series of papers. In this paper, a complete classification is given of edge-transitive Cayley graphs of square-free order with valency less than the smallest prime divisor of the order. This leads to new constructions of infinite families of both arc-regular Cayley graphs and edge-regular Cayley graphs (so half-transitive). Also, as by-products, it is proved that, for any given positive integers $$k,s\ge 1$$ and $$m,n\ge 2$$ , there are infinitely many arc-regular normal circulants of valency 2k and order a product of s primes, and there are infinitely many edge-regular normal metacirculants of valency 2m and order a product of n primes; such arc-regular and edge-regular examples are also specifically constructed.

Formal Duality in Finite Cyclic Groups

The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there are no primitive formally dual pairs besides the trivial one and the TITO configuration. This conjecture has been verified for cyclic groups of prime power order, as well as of square-free order. In this paper, we will confirm the conjecture for other classes of cyclic groups, namely almost all cyclic groups of order a product of two prime powers, with finitely many exceptions for each pair of primes, or whose order $N$ satisfies $p\mid\!\mid N$, where $p$ a prime satisfying the so-called self-conjugacy property with respect to $N$. For the above proofs, various tools were needed: the field descent method, used chiefly for the circulant Hadamard conjecture, the techniques of Coven & Meyerowitz for sets that tile $\mathbb{Z}$ or $\mathbb{Z}_N$ by translations, dubbed herein as the polynomial method, as well as basic number theory of cyclotomic fields, especially the splitting of primes in a given cyclotomic extension.

平方自由阶素数度2-弧正则图

平方自由阶素数度2-弧正则图

On prime-valent symmetric graphs of square-free order

Symmetric graphs of valencies 3, 4 and 5 and square-free order have been classified in the literature. In this paper, we will present a complete classification of symmetric graphs of square-free order and any prime valency which admit a soluble arc-transitive group, and a complete classification of 7-valent symmetric graphs of square-free order.